Asymptotic behavior of weak solutions to the damped Navier-Stokes equations

Whether or not weak solutions of classical Navier-Stokes equations decay to zero in L-2 as time tends to infinity was posed by Leray in his pioneering paper [12,13] and has been well solved (see [15] for instance). This paper examines the three dimensional Navier-Stokes equations with damping term vertical bar u vertical bar(beta-1)u and proves that the weak solutions decay to zero in L-2 as time tends to infinity for any beta >= 1, by developing local-in-space estimate for weak solutions. (C) 2019 Elsevier Inc. All rights reserved.